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Full text of "Analytical Mechanics"

WORK
171
placement and F the force on the piston then the wort
done is                         ,JJ7        _ ,
dW = - F dx
= - pA dx = - p dv,
where p is the pressure in the fluid, A the area of the piston and dv the change in the volume of the fluid. Therefore th< total work done in compressing the fluid from a volume v to a volume t?2 is
W= -  I "p<fo.                             (!'
Jfl!
When the law connecting p and v is given the work dom
in compressing or expanding a fluid can be found by carry
ing   out    the   integration
indicated in equation  (1).
During expansion, however,
the  displacement   has  the
same direction as the force
which causes the expansion;
therefore the sign before the
integral is positive.
141.   Representation of the Work Done in the PV-Dia-gram. When the  volume of the  expanding fluid is plotted as abscissa and the
pressure as ordinate, a curve is obtained, which repre sents, graphically, the law connecting p and v. Such representation is called a PV-diagram. It is evident fror equation (1) that the area-bounded by the curve, the *;-axi! and the two vertical lines whose equations are v = DI an v = tb, represents the work done in compressing the flui from Vi to V*.
142.   Isothermal Compression of a Gas.  If a gas is coir pressed without changing its temperature the compressio
FIG. 96.